The Poisson process

There exist two alternative ways to study a sequence of event arrivals. First we can consider the sequence of arrival dates or equivalently the sequence of durations

Y1r…, Yn,… between consecutive events. Secondly, we can introduce the counting process [N(t), t varying], which counts the number of events observed between 0 and t. The counting process is a jump process, whose jumps of unitary size occur at each arrival date. It is equivalent to know the sequence of durations or the path of the counting process.

The Poisson process is obtained by imposing the following two conditions:

This result explains why basic duration models are based on exponential distri­butions, whereas basic models for count data are based on Poisson distributions, and how these specifications are related (see Cameron and Trivedi, Chapter 15, in this volume).

Similarly, a more complex dynamics can be obtained by relaxing the assump­tion that either the successive durations, or the increments of the counting pro­cess are independent. In the following subsections we introduce the temporal dependence duration sequences.